Answer :
Let's go through the process step-by-step to figure out how many pounds of each type of nut the customer buys.
### Equation Setup:
1. Variables:
- Let [tex]\( x \)[/tex] be the number of pounds of cashews.
- Let [tex]\( y \)[/tex] be the number of pounds of almonds.
- Let [tex]\( z \)[/tex] be the number of pounds of walnuts.
2. Equations based on the problem:
- Equation 1: From "the customer buys 2 more pounds of walnuts than cashews," we get:
[tex]\[
z = x + 2
\][/tex]
- Equation 2: From the total cost of the nuts being $118:
[tex]\[
7y + 10x + 12z = 118
\][/tex]
- Equation 3: From the total weight of the mixed nuts being 12 pounds:
[tex]\[
x + y + z = 12
\][/tex]
### Solving the Equations:
1. Substitute [tex]\( z \)[/tex] from Equation 1:
Use [tex]\( z = x + 2 \)[/tex] and substitute it in Equations 2 and 3.
2. Substitute in Equation 3:
[tex]\[
x + y + (x + 2) = 12
\][/tex]
Simplify:
[tex]\[
2x + y + 2 = 12
\][/tex]
Subtract 2 from both sides:
[tex]\[
2x + y = 10
\][/tex]
So:
[tex]\[
y = 10 - 2x \tag{Equation 4}
\][/tex]
3. Substitute in Equation 2:
[tex]\[
7y + 10x + 12(x + 2) = 118
\][/tex]
Substitute [tex]\( y = 10 - 2x \)[/tex] from Equation 4:
[tex]\[
7(10 - 2x) + 10x + 12(x + 2) = 118
\][/tex]
Simplify:
[tex]\[
70 - 14x + 10x + 12x + 24 = 118
\][/tex]
Combine terms:
[tex]\[
70 + 24 + 8x = 118
\][/tex]
Simplify:
[tex]\[
94 + 8x = 118
\][/tex]
Subtract 94 from both sides:
[tex]\[
8x = 24
\][/tex]
Divide by 8:
[tex]\[
x = 3
\][/tex]
4. Find [tex]\( y \)[/tex] and [tex]\( z \)[/tex]:
Using [tex]\( x = 3 \)[/tex] in Equation 4:
[tex]\[
y = 10 - 2(3)
\][/tex]
[tex]\[
y = 10 - 6
\][/tex]
[tex]\[
y = 4
\][/tex]
Use [tex]\( x = 3 \)[/tex] in Equation 1 to find [tex]\( z \)[/tex]:
[tex]\[
z = x + 2
\][/tex]
[tex]\[
z = 3 + 2
\][/tex]
[tex]\[
z = 5
\][/tex]
### Conclusion:
- The customer buys 3 pounds of cashews ([tex]\(x = 3\)[/tex]).
- The customer buys 4 pounds of almonds ([tex]\(y = 4\)[/tex]).
- The customer buys 5 pounds of walnuts ([tex]\(z = 5\)[/tex]).
Therefore, a possible interpretation of the situation is that the customer buys 2 more pounds of walnuts than almonds and 1 more pound of almonds than cashews.
### Equation Setup:
1. Variables:
- Let [tex]\( x \)[/tex] be the number of pounds of cashews.
- Let [tex]\( y \)[/tex] be the number of pounds of almonds.
- Let [tex]\( z \)[/tex] be the number of pounds of walnuts.
2. Equations based on the problem:
- Equation 1: From "the customer buys 2 more pounds of walnuts than cashews," we get:
[tex]\[
z = x + 2
\][/tex]
- Equation 2: From the total cost of the nuts being $118:
[tex]\[
7y + 10x + 12z = 118
\][/tex]
- Equation 3: From the total weight of the mixed nuts being 12 pounds:
[tex]\[
x + y + z = 12
\][/tex]
### Solving the Equations:
1. Substitute [tex]\( z \)[/tex] from Equation 1:
Use [tex]\( z = x + 2 \)[/tex] and substitute it in Equations 2 and 3.
2. Substitute in Equation 3:
[tex]\[
x + y + (x + 2) = 12
\][/tex]
Simplify:
[tex]\[
2x + y + 2 = 12
\][/tex]
Subtract 2 from both sides:
[tex]\[
2x + y = 10
\][/tex]
So:
[tex]\[
y = 10 - 2x \tag{Equation 4}
\][/tex]
3. Substitute in Equation 2:
[tex]\[
7y + 10x + 12(x + 2) = 118
\][/tex]
Substitute [tex]\( y = 10 - 2x \)[/tex] from Equation 4:
[tex]\[
7(10 - 2x) + 10x + 12(x + 2) = 118
\][/tex]
Simplify:
[tex]\[
70 - 14x + 10x + 12x + 24 = 118
\][/tex]
Combine terms:
[tex]\[
70 + 24 + 8x = 118
\][/tex]
Simplify:
[tex]\[
94 + 8x = 118
\][/tex]
Subtract 94 from both sides:
[tex]\[
8x = 24
\][/tex]
Divide by 8:
[tex]\[
x = 3
\][/tex]
4. Find [tex]\( y \)[/tex] and [tex]\( z \)[/tex]:
Using [tex]\( x = 3 \)[/tex] in Equation 4:
[tex]\[
y = 10 - 2(3)
\][/tex]
[tex]\[
y = 10 - 6
\][/tex]
[tex]\[
y = 4
\][/tex]
Use [tex]\( x = 3 \)[/tex] in Equation 1 to find [tex]\( z \)[/tex]:
[tex]\[
z = x + 2
\][/tex]
[tex]\[
z = 3 + 2
\][/tex]
[tex]\[
z = 5
\][/tex]
### Conclusion:
- The customer buys 3 pounds of cashews ([tex]\(x = 3\)[/tex]).
- The customer buys 4 pounds of almonds ([tex]\(y = 4\)[/tex]).
- The customer buys 5 pounds of walnuts ([tex]\(z = 5\)[/tex]).
Therefore, a possible interpretation of the situation is that the customer buys 2 more pounds of walnuts than almonds and 1 more pound of almonds than cashews.